Handbook of computability and complexity in analysis:
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| Other Authors: | , |
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| Format: | Book |
| Language: | English |
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Cham, Switzerland
Springer
[2021]
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| Series: | Theory and applications of computability
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| Online Access: | Inhaltsverzeichnis |
| Item Description: | Auf dem äußeren Buchdeckel: In cooperation with the association Computability in Europe |
| Physical Description: | xxv, 427 Seiten |
| ISBN: | 9783030592332 |
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| 245 | 1 | 0 | |a Handbook of computability and complexity in analysis |c Vasco Brattka, Peter Hertling, editors |
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| 490 | 0 | |a Theory and applications of computability | |
| 500 | |a Auf dem äußeren Buchdeckel: In cooperation with the association Computability in Europe | ||
| 650 | 4 | |a Theory of Computation | |
| 650 | 4 | |a Mathematics of Computing | |
| 650 | 4 | |a Mathematical Logic and Foundations | |
| 650 | 4 | |a Computers | |
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| adam_text | Contents Preface.................................................................................................................... vii Contents.................................................................................................................. xv List of Contributors.............................................................................................. xxiii Part I Computability in Analysis 1 Computability of Real Numbers............................................................... 3 Robert Rettinger and Xizhong Zheng 1.1 Introduction..................................................................................... 3 1.2 Computable Real Numbers........................................................... 5 1.3 A Finite Hierarchy of Computably Approximable Real Numbers . 7 1A Differences of C.E. Real Numbers............................................... 11 1.5 Divergence Bounded Computable Real Numbers....................... 16 1.6 Turing Degrees of Computably Approximable Real Numbers .... 22 References........................................................................................................ 26 2 Computability of Subsets of Metric Spaces............................................ Zvonko Iljazović and Takayuki Kihara 2.1 Introduction........................................................................................... 2.2 Computable Subsets of Euclidean Space.......................................... 2.3 Computable Metric Spaces ................................................................. 2.3.1 Computable Compact and Closed
Sets.............................. 2.4 Noncomputability of Points in Co-C.E. Closed Sets......................... 2.4.1 Basis Theorems in Computability Theory ........................ 2.4.2 Basis Theorems in Computable Analysis.......................... 2.5 Represented Spaces and Uniform Computability............................. 2.5.1 Represented Hyperspaces.................................................... 2.5.2 Represented Function Spaces.............................................. 2.5.3 Borel Codes.......................................................................... 2.6 Computability of Connectedness Notions......................................... 29 29 30 32 32 36 36 37 39 40 42 44 46 XV
xvi 3 4 Contents 2.6.1 Effective Connectivity Properties ...................................... 2.6.2 Computable Graph Theorem.............................................. 2.6.3 Degrees of Difficulty........................................................... 2.7 Classification of Polish Spaces........................................................ 2.7.1 Borel Isomorphism Theorem.............................................. 2.7.2 Continuous Degree Theory ................................................ 2.7.3 Computable Aspects of InfiniteDimensionality ................ 2.8 Computability of Semicomputable Sets........................................... 2.8.1 Semicomputable Chainable and Circularly Chainable Continua............................................................... 53 2.8.2 Semicomputable Manifolds.............................................. 2.8.3 Inner Approximation.......................................................... 2.8.4 Density of Computable Points in Semicomputable Sets .. 2.9 Computable Images of a Segment.................................................... 2.10 Computability Structures.................................................................. References.................................................................................................... 46 47 48 49 49 50 51 52 Computability of Differential Equations............................................... Daniel S. Graça and Ning Zhong 3.1 Introduction....................................................................................... 3.2 Computability of the Solutions of
Ordinary Differential Equations 3.2.1 Computability over Compact Sets..................................... 3.2.2 Computability over Non-compact Sets............................. 3.3 Computational Complexity of the Solutions of Ordinary Differential Equations .......................................................... 77 3.3.1 Results for Compact Sets................................................... 3.3.2 Results for Non-compact Sets........................................... 3.4 Computability of Qualitative Behaviors of Ordinary Differential Equations............................................................................... 85 3.5 Computability of Partial Differential Equations............................. 3.6 Some Open Problems....................................................................... References................................................................................................... 71 55 58 58 61 62 65 71 73 73 74 78 82 89 94 94 Computable Complex Analysis.............................................................. 101 Valentin V. Andreev and Timothy H. McNichoIl 4.1 Introduction....................................................................................... 101 4.2 Preliminaries from Complex Analysis............................................. 104 4.3 Computability on the Complex Plane and Extended Complex Plane 106 4.3.1 Computability of Differentiation, Computably Analytic Functions........................................................................... 109 4.3.2 Series
...................................................................................109 4.3.3 Zeros.................................................................................... 110 4.3.4 Open Mapping..................................................................... Ill 4.4 Computable Conformal Mapping of Simply Connected Domains .111 4.4.1 Classical Background.......................................................... Ill 4.4.2 Computability Results ........................................................ 112
Contents xvii 4.4.3 Complexity Results................................................................115 Boundary Extensions............................................................................ 116 4.5.1 Classical Background............................................................. 116 4.5.2 Computability Results ...........................................................117 4.6 Harmonic Functions.............................................................................. 120 4.6.1 Classical Background............................................................. 120 4.6.2 Computability Results ...........................................................123 4.7 Conformal Mapping of Multiply Connected Domains...................... 125 4.7.1 Classical Background.............................................................125 4.7.2 Computability Results ........................................................... 130 4.8 Infinite Products.................................................................................... 131 4.8.1 Classical Background............................................................. 131 4.8.2 Computability Results ...........................................................132 4.9 Constants................................................................................................ 133 4.9.1 Classical Background............................................................. 133 4.9.2 Computability Results ........................................................... 134 4.9.3 Complexity
Results................................................................. 134 4.10 Open Problems...................................................................................... 134 4.10.1 The Hayman-Wu Constant..................................................... 135 4.10.2 Parameterized Complexity Riemann Mapping Theorem .135 4.10.3 Computability ConformalMapping onto Infinitely Connected Domains ............................................................. 135 References......................................................................................................... 136 4.5 Part II Complexity, Dynamics, and Randomness 5 Computable Geometric Complex Analysis and Complex Dynamics.. 143 Cristóbal Rojas, Michael Yampolsky 5.1 Introduction............................................................................................ 143 5.2 Required Computability Notions ........................................................ 144 5.2.1 Computable Metric Spaces.................................................... 144 5.2.2 Computability of Probability Measures .............................. 146 5.2.3 Time Complexity of a Problem............................................147 5.2.4 Computational Complexity of Two-Dimensional Images . 148 5.3 Computability and Complexity of Conformal Mappings................. 149 5.4 Computable Carathéodory Theory.......................................................150 5.4.1 Carathéodory Extension Theorem........................................ 151 5.4.2 Computational Representation of Prime Ends.................... 153 5.4.3
Structure of a Computable Metric Space on the Carathéodory Compactification............................ 153 5.4.4 Moduli of Locally Connected Domains.............................. 154 5.4.5 Computable Carathéodory Theory ...................................... 155 5.5 Computability in Complex Dynamics: Julia Sets.............................. 156 5.5.1 Basic Properties of Julia Sets................................................ 156
xviii Contents Occurrence of Siegel Disks and Cremer Points in the Quadratic Family...................................................... 160 5.5.3 Computability of Julia Sets .................................................... 163 5.5.4 Computational Complexity of Julia Sets............................... 164 5.5.5 Computing Julia Sets in Statistical Terms..............................166 5.5.6 Applications of Computable Carathéodory Theory to Julia Sets: External Rays and Their Impressions.............. 167 5.5.7 On the Computability of the Mandelbrot Set........................ 168 References............................................................................................................ 170 5.5.2 6 A Survey on Analog Models of Computation ......................................... 173 Olivier Bournez and Amaury Pouly 6.1 Introduction .............................................................................................. 173 6.2 Various Analog Machines and Models.................................................. 176 6.2.1 Historical Accounts.................................................................. 176 6.2.2 Differential Analyzers.............................................................. 177 6.2.3 Neural Networks and Deep Learning Models ..................... 178 6.2.4 Models from Verification........................................................ 180 6.2.5 Blum-Shub-Smale’s Model .................................................... 181 6.2.6 Natural Computing.................................................................. 182 6.2.7
Solving Various Problems Using Dynamical Systems ... 183 6.2.8 Distributed Computing .......................................................... 185 6.2.9 Black Hole Models...................................................................187 6.2.10 Spatial Models........................................................................... 188 6.2.11 Various Other Models............................................................... 190 6.3 Dynamical Systems and Computations ................................................. 191 6.3.1 Arbitrary Versus Rational/Computable Reals....................... 191 6.3.2 Static Undecidability................................................................ 191 6.3.3 Dynamic Undecidability ........................................................ 192 6.4 Philosophical, Mathematical and Physics-Related Aspects ............ 194 6.4.1 Mathematical Models Versus Systems................................. 194 6.4.2 Church-Turing Thesis and Variants ..................................... 195 6.4.3 Are Analog Systems Capable of Hypercomputations? ... 196 6.4.4 Can Analog Machines Compute Faster?................................197 6.4.5 Some Philosophical Aspects ................................................. 198 6.5 Theory of Analog Systems ...................................................................199 6.5.1 Generic Formalizations of Analog Computations............. 199 6.5.2 R-recursion Theory................................................................ 201 6.5.3 Analog Automata Theories ..................................................
202 6.6 Analyzing the Power and Limitations of Analog Computations ... 203 6.6.1 Neural Networks ...................................................................203 6.6.2 Physical Oracles ..................................................................... 203 6.6.3 On the Effect of Noise on Computations.............................205 6.6.4 Complexity Theories for Analog Computations ................. 205 6.6.5 Chemical Reaction Networks .............................................. 206
Contents xix Computations by Polynomial Ordinary Differential Equations .... 207 6.7.1 GPAC and Polynomial Ordinary Differential Equations .207 6.7.2 GPAC Generable Functions.................................................. 208 6.7.3 GPAC Computability ............................................................ 208 References......................................................................................................... 211 6.7 7 Computable Measure Theory and Algorithmic Randomness.............227 Mathieu Hoyrup and Jason Rute 7.1 Introduction............................................................................................ 227 7.2 Computable Measure Theory .............................................................. 228 7.2.1 Background from Computable Analysis.............................. 228 7.2.2 Framework.............................................................................. 229 7.2.3 Results in Computable Measure and Probability Theory . 237 7.3 Algorithmic Randomness...................................................................... 242 7.3.1 Effective Null Sets.................................................................. 243 7.3.2 Effective Convergence Theorems ........................................ 247 7.3.3 Randomness Preservation...................................................... 249 7.3.4 Product Spaces........................................................................ 251 7.4 Pointwise Computable Measure Theory.............................................. 254 7.4.1 Effective
Tightness................................................................ 255 7.4.2 Effective Egorov’s Theorem.................................................. 256 7.4.3 Effective Lusin Theorem...................................................... 256 7.4.4 Effective Absolute Continuity.............................................. 257 7.4.5 Properties of Layerwise Computable Functions................. 258 7.4.6 Randomness via Encoding................................................... 260 7.4.7 Recovering a Distribution from a Sample........................... 262 References......................................................................................................... 264 8 Algorithmic Fractal Dimensions in Geometric Measure Theory.........271 Jack H. Lutz and Elvira Mayordomo 8.1 Introduction............................................................................................ 271 8.2 Algorithmic Information in EuclideanSpaces...................................... 274 8.3 Algorithmic Dimensions...................................................................... 275 8.3.1 Dimensions of Points........................................................... 275 8.3.2 The Correspondence Principle.............................................. 277 8.3.3 Self-Similar Fractals..............................................................278 8.3.4 Dimension Level Sets........................................................... 281 8.3.5 Dimensions of Points on Lines ............................................ 282 8.4 Mutual and Conditional Dimensions
................................................... 284 8.4.1 Mutual Dimensions................................................................284 8.4.2 Data Processing Inequalities ................................................ 285 8.4.3 Conditional Dimensions........................................................ 286 8.5 Algorithmic Discovery of New ClassicalTheorems............................ 288 8.5.1 The Point-to-Set Principle....................................................288 8.5.2 Plane Kakeya Sets.................................................................. 289 8.5.3 Intersections and Products of Fractals.................................. 291
Contents XX 8.5.4 Generalized Furstenberg Sets............................................. 293 Research Directions.......................................................................... 293 8.6.1 Beyond Self-Similarity ...................................................... 293 8.6.2 Beyond Euclidean Spaces.................................................... 295 8.6.3 Beyond Computability........................................................ 295 8.6.4 Beyond Fractals................................................................... 296 References..................................................................................................... 298 8.6 Part III Constructivity, Logic, and Descriptive Complexity 9 Admissibly Represented Spaces and Qcb-Spaces..................................... 305 Matthias Schröder 9.1 Introduction.........................................................................................305 9.2 Represented Spaces ........................................................................... 306 9.2.1 Representations................................................................... 306 9.2.2 The Baire Space................................................................... 307 9.2.3 Computability on the Baire Space...................................... 308 9.2.4 Represented Spaces............................................................. 310 9.2.5 Computable Elements of Represented Spaces................... 310 9.2.6 Computable Realizability.................................................... 310 9.2.7 The Cauchy Representation of the
Real Numbers..............312 9.2.8 The Signed-Digit Representation........................................ 312 9.2.9 Continuous Realizability .................................................... 313 9.2.10 Reducibility and Equivalence of Representations.............. 314 9.2.11 The Category of Represented Spaces................................. 315 9.2.12 Closure Properties of Represented Spaces......................... 315 9.2.13 Multivalued Functions ........................................................ 317 9.2.14 Multirepresentations............................................................ 318 9.3 Admissible Representations of Topological Spaces.......................... 319 9.3.1 The Topology of a Represented Space............................... 319 9.3.2 Sequential Topological Spaces.............................................320 9.3.3 Sequentialisation of Topological Spaces..............................320 9.3.4 Topological Admissibility .................................................. 321 9.3.5 Examples of Admissible Representations ........................... 323 9.3.6 The Admissibility Notion of Kreitz and Weihrauch.........324 9.3.7 Constructing Admissible Representations......................... 325 9.3.8 Pseudobases........................................................................ 326 9.4 Qcb-Spaces......................................................................................... 327 9.4.1 Qcb-Spaces .........................................................................327 9.4.2 Operators on Qcb-
Spaces.................................................... 327 9.4.3 Powerspaces in QCBq......................................................... 329 9.4.4 Qcb-Spaces and Admissibility............................................ 332 9.4.5 Effectively Admissible Representations............................. 334 9.4.6 Effective Qcb-Spaces.......................................................... 335 9.4.7 Basic Computable Functions on Effective Qcb-Spaces... 337
Contents xxi 9.4.8 Effective Hausdorff Spaces .................................................. 338 9.4.9 Quasi-normal Qcb-Spaces.................................................... 338 9.5 Relationship to Other Relevant Categories ....................................... 340 9.5.1 Represented Spaces................................................................ 340 9.5.2 Equilogical Spaces ................................................................ 340 9.5.3 LimitSpaces .......................................................................... 341 9.5.4 Cartesian Closed Subcategories of Topological Spaces .. 343 References......................................................................................................... 344 10 Bishop-Style Constructive Reverse Mathematics .................................... 347 Hannes Diener and Hajime Ishihara 10.1 Introduction........................................................................................... 347 10.2 Preliminaries......................................................................................... 348 10.3 Omniscience Principles: LPO, WLPO, and LLPO........................... 352 10.4 Markov’s Principle and Related Principles: MP, WMP, and MPV .354 10.5 A Continuity Principle, the Fan Theorem, and Church’s Thesis ... 356 10.6 The Boundedness Principle: BD-N..................................................... 359 10.7 Relationships Between Principles....................................................... 361 10.8 Separation
Techniques......................................................................... 361 References......................................................................................................... 363 11 Weihrauch Complexity in Computable Analysis...................................... 367 Vasco Brattka, Guido Gherardi and Arno Pauly 11.1 The Algebra of Problems...................................................................... 367 11.2 Represented Spaces .............................................................................. 370 11.3 The Weihrauch Lattice.......................................................................... 374 11.4 Algebraic and Topological Properties.................................................. 377 11.5 Completeness, Compositionand Implication ..................................... 380 11.6 Limits and Jumps.................................................................................. 382 11.7 Choice.................................................................................................... 387 11.7.1 Composition and Non-determinism ................................... 390 11.7.2 Choice on Natural Numbers.................................................. 391 11.7.3 Choice on the Cantor Space.................................................. 392 11.7.4 Choice on Euclidean Space.................................................. 394 11.7.5 Choice on the Baire Space.....................................................396 11.7.6 Jumps of Choice.................................................................... 396 11.7.7
All-or-Unique Choice............................................................ 398 11.8 Classifications........................................................................................ 399 11.9 Relations to Other Theories.................................................................. 405 11.9.1 Linear Logic............................................................................ 405 11.9.2 Medvedev Lattice, Many-One and Turing Semilattices .. 406 11.9.3 Reverse Mathematics ............................................................ 407 11.9.4 Constructive Reverse Mathematics...................................... 408 11.9.5 Other Reducibilities .............................................................. 409 11.9.6 Descriptive Set Theory.......................................................... 410 11.9.7 Other Models of Computability............................................ 411
xxii Contents References........................................................................................................... 412 Index............................................................................................................................ 419
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Contents Preface. vii Contents. xv List of Contributors. xxiii Part I Computability in Analysis 1 Computability of Real Numbers. 3 Robert Rettinger and Xizhong Zheng 1.1 Introduction. 3 1.2 Computable Real Numbers. 5 1.3 A Finite Hierarchy of Computably Approximable Real Numbers . 7 1A Differences of C.E. Real Numbers. 11 1.5 Divergence Bounded Computable Real Numbers. 16 1.6 Turing Degrees of Computably Approximable Real Numbers . 22 References. 26 2 Computability of Subsets of Metric Spaces. Zvonko Iljazović and Takayuki Kihara 2.1 Introduction. 2.2 Computable Subsets of Euclidean Space. 2.3 Computable Metric Spaces . 2.3.1 Computable Compact and Closed
Sets. 2.4 Noncomputability of Points in Co-C.E. Closed Sets. 2.4.1 Basis Theorems in Computability Theory . 2.4.2 Basis Theorems in Computable Analysis. 2.5 Represented Spaces and Uniform Computability. 2.5.1 Represented Hyperspaces. 2.5.2 Represented Function Spaces. 2.5.3 Borel Codes. 2.6 Computability of Connectedness Notions. 29 29 30 32 32 36 36 37 39 40 42 44 46 XV
xvi 3 4 Contents 2.6.1 Effective Connectivity Properties . 2.6.2 Computable Graph Theorem. 2.6.3 Degrees of Difficulty. 2.7 Classification of Polish Spaces. 2.7.1 Borel Isomorphism Theorem. 2.7.2 Continuous Degree Theory . 2.7.3 Computable Aspects of InfiniteDimensionality . 2.8 Computability of Semicomputable Sets. 2.8.1 Semicomputable Chainable and Circularly Chainable Continua. 53 2.8.2 Semicomputable Manifolds. 2.8.3 Inner Approximation. 2.8.4 Density of Computable Points in Semicomputable Sets . 2.9 Computable Images of a Segment. 2.10 Computability Structures. References. 46 47 48 49 49 50 51 52 Computability of Differential Equations. Daniel S. Graça and Ning Zhong 3.1 Introduction. 3.2 Computability of the Solutions of
Ordinary Differential Equations 3.2.1 Computability over Compact Sets. 3.2.2 Computability over Non-compact Sets. 3.3 Computational Complexity of the Solutions of Ordinary Differential Equations . 77 3.3.1 Results for Compact Sets. 3.3.2 Results for Non-compact Sets. 3.4 Computability of Qualitative Behaviors of Ordinary Differential Equations. 85 3.5 Computability of Partial Differential Equations. 3.6 Some Open Problems. References. 71 55 58 58 61 62 65 71 73 73 74 78 82 89 94 94 Computable Complex Analysis. 101 Valentin V. Andreev and Timothy H. McNichoIl 4.1 Introduction. 101 4.2 Preliminaries from Complex Analysis. 104 4.3 Computability on the Complex Plane and Extended Complex Plane 106 4.3.1 Computability of Differentiation, Computably Analytic Functions. 109 4.3.2 Series
.109 4.3.3 Zeros. 110 4.3.4 Open Mapping. Ill 4.4 Computable Conformal Mapping of Simply Connected Domains .111 4.4.1 Classical Background. Ill 4.4.2 Computability Results . 112
Contents xvii 4.4.3 Complexity Results.115 Boundary Extensions. 116 4.5.1 Classical Background. 116 4.5.2 Computability Results .117 4.6 Harmonic Functions. 120 4.6.1 Classical Background. 120 4.6.2 Computability Results .123 4.7 Conformal Mapping of Multiply Connected Domains. 125 4.7.1 Classical Background.125 4.7.2 Computability Results . 130 4.8 Infinite Products. 131 4.8.1 Classical Background. 131 4.8.2 Computability Results .132 4.9 Constants. 133 4.9.1 Classical Background. 133 4.9.2 Computability Results . 134 4.9.3 Complexity
Results. 134 4.10 Open Problems. 134 4.10.1 The Hayman-Wu Constant. 135 4.10.2 Parameterized Complexity Riemann Mapping Theorem .135 4.10.3 Computability ConformalMapping onto Infinitely Connected Domains . 135 References. 136 4.5 Part II Complexity, Dynamics, and Randomness 5 Computable Geometric Complex Analysis and Complex Dynamics. 143 Cristóbal Rojas, Michael Yampolsky 5.1 Introduction. 143 5.2 Required Computability Notions . 144 5.2.1 Computable Metric Spaces. 144 5.2.2 Computability of Probability Measures . 146 5.2.3 Time Complexity of a Problem.147 5.2.4 Computational Complexity of Two-Dimensional Images . 148 5.3 Computability and Complexity of Conformal Mappings. 149 5.4 Computable Carathéodory Theory.150 5.4.1 Carathéodory Extension Theorem. 151 5.4.2 Computational Representation of Prime Ends. 153 5.4.3
Structure of a Computable Metric Space on the Carathéodory Compactification. 153 5.4.4 Moduli of Locally Connected Domains. 154 5.4.5 Computable Carathéodory Theory . 155 5.5 Computability in Complex Dynamics: Julia Sets. 156 5.5.1 Basic Properties of Julia Sets. 156
xviii Contents Occurrence of Siegel Disks and Cremer Points in the Quadratic Family. 160 5.5.3 Computability of Julia Sets . 163 5.5.4 Computational Complexity of Julia Sets. 164 5.5.5 Computing Julia Sets in Statistical Terms.166 5.5.6 Applications of Computable Carathéodory Theory to Julia Sets: External Rays and Their Impressions. 167 5.5.7 On the Computability of the Mandelbrot Set. 168 References. 170 5.5.2 6 A Survey on Analog Models of Computation . 173 Olivier Bournez and Amaury Pouly 6.1 Introduction . 173 6.2 Various Analog Machines and Models. 176 6.2.1 Historical Accounts. 176 6.2.2 Differential Analyzers. 177 6.2.3 Neural Networks and Deep Learning Models . 178 6.2.4 Models from Verification. 180 6.2.5 Blum-Shub-Smale’s Model . 181 6.2.6 Natural Computing. 182 6.2.7
Solving Various Problems Using Dynamical Systems . 183 6.2.8 Distributed Computing . 185 6.2.9 Black Hole Models.187 6.2.10 Spatial Models. 188 6.2.11 Various Other Models. 190 6.3 Dynamical Systems and Computations . 191 6.3.1 Arbitrary Versus Rational/Computable Reals. 191 6.3.2 Static Undecidability. 191 6.3.3 Dynamic Undecidability . 192 6.4 Philosophical, Mathematical and Physics-Related Aspects . 194 6.4.1 Mathematical Models Versus Systems. 194 6.4.2 Church-Turing Thesis and Variants . 195 6.4.3 Are Analog Systems Capable of Hypercomputations? . 196 6.4.4 Can Analog Machines Compute Faster?.197 6.4.5 Some Philosophical Aspects . 198 6.5 Theory of Analog Systems .199 6.5.1 Generic Formalizations of Analog Computations. 199 6.5.2 R-recursion Theory. 201 6.5.3 Analog Automata Theories .
202 6.6 Analyzing the Power and Limitations of Analog Computations . 203 6.6.1 Neural Networks .203 6.6.2 Physical Oracles . 203 6.6.3 On the Effect of Noise on Computations.205 6.6.4 Complexity Theories for Analog Computations . 205 6.6.5 Chemical Reaction Networks . 206
Contents xix Computations by Polynomial Ordinary Differential Equations . 207 6.7.1 GPAC and Polynomial Ordinary Differential Equations .207 6.7.2 GPAC Generable Functions. 208 6.7.3 GPAC Computability . 208 References. 211 6.7 7 Computable Measure Theory and Algorithmic Randomness.227 Mathieu Hoyrup and Jason Rute 7.1 Introduction. 227 7.2 Computable Measure Theory . 228 7.2.1 Background from Computable Analysis. 228 7.2.2 Framework. 229 7.2.3 Results in Computable Measure and Probability Theory . 237 7.3 Algorithmic Randomness. 242 7.3.1 Effective Null Sets. 243 7.3.2 Effective Convergence Theorems . 247 7.3.3 Randomness Preservation. 249 7.3.4 Product Spaces. 251 7.4 Pointwise Computable Measure Theory. 254 7.4.1 Effective
Tightness. 255 7.4.2 Effective Egorov’s Theorem. 256 7.4.3 Effective Lusin Theorem. 256 7.4.4 Effective Absolute Continuity. 257 7.4.5 Properties of Layerwise Computable Functions. 258 7.4.6 Randomness via Encoding. 260 7.4.7 Recovering a Distribution from a Sample. 262 References. 264 8 Algorithmic Fractal Dimensions in Geometric Measure Theory.271 Jack H. Lutz and Elvira Mayordomo 8.1 Introduction. 271 8.2 Algorithmic Information in EuclideanSpaces. 274 8.3 Algorithmic Dimensions. 275 8.3.1 Dimensions of Points. 275 8.3.2 The Correspondence Principle. 277 8.3.3 Self-Similar Fractals.278 8.3.4 Dimension Level Sets. 281 8.3.5 Dimensions of Points on Lines . 282 8.4 Mutual and Conditional Dimensions
. 284 8.4.1 Mutual Dimensions.284 8.4.2 Data Processing Inequalities . 285 8.4.3 Conditional Dimensions. 286 8.5 Algorithmic Discovery of New ClassicalTheorems. 288 8.5.1 The Point-to-Set Principle.288 8.5.2 Plane Kakeya Sets. 289 8.5.3 Intersections and Products of Fractals. 291
Contents XX 8.5.4 Generalized Furstenberg Sets. 293 Research Directions. 293 8.6.1 Beyond Self-Similarity . 293 8.6.2 Beyond Euclidean Spaces. 295 8.6.3 Beyond Computability. 295 8.6.4 Beyond Fractals. 296 References. 298 8.6 Part III Constructivity, Logic, and Descriptive Complexity 9 Admissibly Represented Spaces and Qcb-Spaces. 305 Matthias Schröder 9.1 Introduction.305 9.2 Represented Spaces . 306 9.2.1 Representations. 306 9.2.2 The Baire Space. 307 9.2.3 Computability on the Baire Space. 308 9.2.4 Represented Spaces. 310 9.2.5 Computable Elements of Represented Spaces. 310 9.2.6 Computable Realizability. 310 9.2.7 The Cauchy Representation of the
Real Numbers.312 9.2.8 The Signed-Digit Representation. 312 9.2.9 Continuous Realizability . 313 9.2.10 Reducibility and Equivalence of Representations. 314 9.2.11 The Category of Represented Spaces. 315 9.2.12 Closure Properties of Represented Spaces. 315 9.2.13 Multivalued Functions . 317 9.2.14 Multirepresentations. 318 9.3 Admissible Representations of Topological Spaces. 319 9.3.1 The Topology of a Represented Space. 319 9.3.2 Sequential Topological Spaces.320 9.3.3 Sequentialisation of Topological Spaces.320 9.3.4 Topological Admissibility . 321 9.3.5 Examples of Admissible Representations . 323 9.3.6 The Admissibility Notion of Kreitz and Weihrauch.324 9.3.7 Constructing Admissible Representations. 325 9.3.8 Pseudobases. 326 9.4 Qcb-Spaces. 327 9.4.1 Qcb-Spaces .327 9.4.2 Operators on Qcb-
Spaces. 327 9.4.3 Powerspaces in QCBq. 329 9.4.4 Qcb-Spaces and Admissibility. 332 9.4.5 Effectively Admissible Representations. 334 9.4.6 Effective Qcb-Spaces. 335 9.4.7 Basic Computable Functions on Effective Qcb-Spaces. 337
Contents xxi 9.4.8 Effective Hausdorff Spaces . 338 9.4.9 Quasi-normal Qcb-Spaces. 338 9.5 Relationship to Other Relevant Categories . 340 9.5.1 Represented Spaces. 340 9.5.2 Equilogical Spaces . 340 9.5.3 LimitSpaces . 341 9.5.4 Cartesian Closed Subcategories of Topological Spaces . 343 References. 344 10 Bishop-Style Constructive Reverse Mathematics . 347 Hannes Diener and Hajime Ishihara 10.1 Introduction. 347 10.2 Preliminaries. 348 10.3 Omniscience Principles: LPO, WLPO, and LLPO. 352 10.4 Markov’s Principle and Related Principles: MP, WMP, and MPV .354 10.5 A Continuity Principle, the Fan Theorem, and Church’s Thesis . 356 10.6 The Boundedness Principle: BD-N. 359 10.7 Relationships Between Principles. 361 10.8 Separation
Techniques. 361 References. 363 11 Weihrauch Complexity in Computable Analysis. 367 Vasco Brattka, Guido Gherardi and Arno Pauly 11.1 The Algebra of Problems. 367 11.2 Represented Spaces . 370 11.3 The Weihrauch Lattice. 374 11.4 Algebraic and Topological Properties. 377 11.5 Completeness, Compositionand Implication . 380 11.6 Limits and Jumps. 382 11.7 Choice. 387 11.7.1 Composition and Non-determinism . 390 11.7.2 Choice on Natural Numbers. 391 11.7.3 Choice on the Cantor Space. 392 11.7.4 Choice on Euclidean Space. 394 11.7.5 Choice on the Baire Space.396 11.7.6 Jumps of Choice. 396 11.7.7
All-or-Unique Choice. 398 11.8 Classifications. 399 11.9 Relations to Other Theories. 405 11.9.1 Linear Logic. 405 11.9.2 Medvedev Lattice, Many-One and Turing Semilattices . 406 11.9.3 Reverse Mathematics . 407 11.9.4 Constructive Reverse Mathematics. 408 11.9.5 Other Reducibilities . 409 11.9.6 Descriptive Set Theory. 410 11.9.7 Other Models of Computability. 411
xxii Contents References. 412 Index. 419 |
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| spelling | Handbook of computability and complexity in analysis Vasco Brattka, Peter Hertling, editors Cham, Switzerland Springer [2021] © 2021 xxv, 427 Seiten txt rdacontent n rdamedia nc rdacarrier Theory and applications of computability Auf dem äußeren Buchdeckel: In cooperation with the association Computability in Europe Theory of Computation Mathematics of Computing Mathematical Logic and Foundations Computers Computer science—Mathematics Mathematical logic Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Computermathematik (DE-588)4788218-9 gnd rswk-swf (DE-588)4143413-4 Aufsatzsammlung gnd-content Computermathematik (DE-588)4788218-9 s Mathematische Logik (DE-588)4037951-6 s DE-604 Brattka, Vasco (DE-588)121389847 edt Hertling, Peter 1965- (DE-588)121342360 edt Erscheint auch als Online-Ausgabe 978-3-030-59234-9 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032990954&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Handbook of computability and complexity in analysis Theory of Computation Mathematics of Computing Mathematical Logic and Foundations Computers Computer science—Mathematics Mathematical logic Mathematische Logik (DE-588)4037951-6 gnd Computermathematik (DE-588)4788218-9 gnd |
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| title | Handbook of computability and complexity in analysis |
| title_auth | Handbook of computability and complexity in analysis |
| title_exact_search | Handbook of computability and complexity in analysis |
| title_exact_search_txtP | Handbook of computability and complexity in analysis |
| title_full | Handbook of computability and complexity in analysis Vasco Brattka, Peter Hertling, editors |
| title_fullStr | Handbook of computability and complexity in analysis Vasco Brattka, Peter Hertling, editors |
| title_full_unstemmed | Handbook of computability and complexity in analysis Vasco Brattka, Peter Hertling, editors |
| title_short | Handbook of computability and complexity in analysis |
| title_sort | handbook of computability and complexity in analysis |
| topic | Theory of Computation Mathematics of Computing Mathematical Logic and Foundations Computers Computer science—Mathematics Mathematical logic Mathematische Logik (DE-588)4037951-6 gnd Computermathematik (DE-588)4788218-9 gnd |
| topic_facet | Theory of Computation Mathematics of Computing Mathematical Logic and Foundations Computers Computer science—Mathematics Mathematical logic Mathematische Logik Computermathematik Aufsatzsammlung |
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